COT 310001
July, 26 2001
Final Exam (110 Total)
Name:_________________
SSN:__________________
1. Find the number of 8letter passwords that can be formed by ordering the letters of
the word DISCRETE
subject to one of the following conditions.
a)
(5pts) Contain a substring EE.
7!
b)
(10 pts) Both E’s come after I .
Here is one possible way to solve this problem. Letter I can occur in 1, 2, 3, 4, 5
and 6 position (from left to right). In accordance to this we can count separately 6
cases.
Case 1: letter I in 1st position. 7!/2! arrangements
Case 2: letter I in the 2nd position. Five choices
(from letters D, S, C, T and R)
for the 1st position times 6!/2! (arrangements of 6 letters with 2 non
distinguishable).
Case 3: letter I in the 3rd position. For the first two positions preceding letter I
we can choose from 5 letters, so there are 5
⋅
4 choices. This should be multiplies
by 5!/2! Of the number of arrangements for the last 3 places.
Case 4: letter I in the 4th position: 5
⋅
4
⋅
3 choices for the preceding 3 positions
and 4!/2! For the last two.
Case 5: letter I in the 5th position: 5
⋅
4
⋅
3
⋅
2 choices for the 4 first positions and
3!/2! Choices for the last three.
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 Spring '09
 Mathematical Induction, 1 k, 2 K, Transitive relation, 5pts

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